Brenden Pappas, Tommy Biershenk, Mike Brisky, John Kernohan and overnight co-leader Tom Kalinowski are tied for fourth at minus 12.
Couch, the winner of last week's Florida Classic, opened with a birdie at the first but then carded three bogeys in a row from the second to drop him down the leaderboard. After rebounding with birdies at Nos. 5 and 7, he eagled the par-5 9th to go out in 2-under 34.
Couch posted two birdies in his first four holes on the back but bogeyed the 17th to drop from the lead. With Pappas in the clubhouse at 14-under, Couch two-putted from 30 feet for birdie at the par-5 closing hole to tie for the top spot.
'I just felt like today I didn't hit it quite as well as I have the last couple of days,' said Couch, who has had at least a share of the lead in each of the first three rounds. 'I hit a lot more stray shots and battled hard to shoot what I shot.'
Couch is looking to become the first player since 1997 to win back-to-back events on the Buy.Com Tour.
Pappas carded a 5-under 67 on Saturday for his portion of the lead. He bogeyed the third hole but quickly reclaimed the lost stroke with a birdie at the fourth. Pappas parred out on the front for a 36.
Back-to-back birdies at Nos. 10 and 11, along with another birdie at No. 13 moved Pappas near the top of the leaderboard. Knowing he needed an eagle at the final hole to get in the last group on Sunday, Pappas knocked his second shot to 18 feet and ran home the eagle effort to gain his position in the final pairing.
'I really wanted to make that putt to get into the final group tomorrow,' said Pappas, who captured the South African PGA Championship in February. 'I could have made a few more birdies on the back nine.'
Hometown favorite Serna, who birdied two of his last four holes for a 68 and sole possession of third, said he would be fine with playing in a high pressure situation in his home country.
'I'm not nervous,' he said. 'I am very hopeful for tomorrow.'
Kalinowski had a chance to stay in the lead for Sunday's final round, but bogeyed 18 to drop down to a tie for fourth.
John Wilson, Steve Haskins and Tim Petrovic round out the top-10 at minus 11.
Click here for full-field scores from the Buy.Com Monterrey Open